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Zbl 0645.10015
Dilcher, Karl
Zeros of Bernoulli, generalized Bernoulli and Euler polynomials. (English)
[J] Mem. Am. Math. Soc. 73, No.386, 94 p. (1988). ISSN 0065-9266

Author's abstract: ``It is shown that the Bernoulli polynomials $B\sb n(z)$, the Euler polynomials $E\sb n(z)$ and the generalized Bernoulli polynomials $B\sp n\sb{\chi}(z)$ associated with certain quadratic characters have no zero inside a parabolic region if n is sufficiently large. Zero-free regions are also found for individual polynomials, and for the partial sums of sine and cosine. The proofs are based on a result on the maximum modulus of the zeros of polynomials related to the $B\sb n(z)$, $E\sb n(z)$ and $B\sp n\sb{\chi}(z)$. Finally, the distribution of the real zeros of $B\sp n\sb{\ell}(z)$ and $E\sb n(z)$ is studied. The results are similar to the known results on the real zeros of $B\sb n(z)$.''
[L.Skula]

MSC 2000:: *11B39 Special numbers, etc.

Keywords: location of zeros; partial sums of power series; Eneström-Kakeya theorem; complementary error function; Bernoulli polynomials; Euler polynomials; generalized Bernoulli polynomials; Zero-free regions

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